Re: Reflexive coequalizers

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

Ah yes, I see now. I had observed that the
underlying diagram of indiscrete graphs was still a
coequalizer -- which unfortunately is to truncate
one's simplicial sets a little too much! Many
thanks for the enlightenment.

Richard


--On 09 October 2006 20:40 George Janelidze wrote:

> Dear Richard,
>
> If we were asking the same question about the category SimplSet of
> simplicial sets instead of Cat, the answer would be obvious since:
>
> (*) The functor Set ---> Set sending X to X^n preserves reflexive
> coequalizers for each natural n. This (very simple) fact should be
> considered as well known since it is involved in one of several well-known
> proofs of monadicity of varieties of universal algebras over Set.
>
> (**) Since SimplSet is a Set-valued functor category, all colimits in it
> reduce to colimits in Set.
>
> For those who are familiar with the standard adjunction between SimplSet and
> Cat, the result for SimplSet will immediately imply the result for Cat
> (since the fundamental category functor being the left adjoint in that
> adjunction preserves all colimits and obviously sends "indiscrete simplicial
> sets" to indiscrete categories).
>
> One could also use various (not too much) truncated simplicial sets instead
> of the simplicial sets of course (e.g. what I once called "precategories").
>
> I mean, I do not remember any reference, but I would not need the explicit
> description of colimits in Cat.
>
> George Janelidze
>
> ----- Original Message -----
> From: "Richard Garner" <rhgg2@hermes.cam.ac.uk>
> To: <categories@mta.ca>
> Sent: Monday, October 09, 2006 12:14 PM
> Subject: categories: Reflexive coequalizers
>
>
>>
>> Dear categorists,
>>
>> I have a proof that the indiscrete category functor
>> Set -> Cat preserves reflexive coequalizers which,
>> although straightfoward, uses the explicit
>> description of colimits in Cat. Is this necessary,
>> or can I deduce the result from general
>> principles?
>>
>> [Also, I'm sure this result must appear somewhere
>> but I can't find a reference for it. If anyone
>> knows of one, I'd be grateful.]
>>
>> Many thanks,
>>
>> Richard
>>
>>
>>
>
>